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Jintong Ren. Optimization algorithms for graph layout problems. Data Structures and Algorithms [cs.DS]. Université d’Angers, 2020. English. �NNT : 2020ANGE0012�. �tel-03178385�
T HÈSE DE DOCTORAT DE
L’UNIVERSITÉ DE ANGERS
COMUEUNIVERSITÉ BRETAGNE LOIRE
ÉCOLE DOCTORALE NO601
Mathématiques et Sciences et Technologies de l’Information et de la Communication Spécialité :Informatique
Par
Jintong REN
Optimization algorithms for graph layout problems
Thèse présentée et soutenue à « LERIA », le « 15 Juillet 2020 » Unité de recherche :
Thèse No :
Rapporteurs avant soutenance :
M. Nicolas Durand Professeur à Ecole Nationale de l’Aviation Civile Mme. Yang Wang Professeur à Northwestern Polytechnical University
Composition du Jury :
Président : M. David Lesaint Professeur à Université d’Angers
Examinateurs : M. Nicolas Durand Professeur à Ecole Nationale de l’Aviation Civile M. Jin-Kao Hao Professeur à Université d’Angers
M. David Lesaint Professeur à Université d’Angers M. Eduardo Rodriguez-Tello Professeur à CINVESTAV, Tamaulipas
Mme. Yang Wang Professeur à Northwestern Polytechnical University Dir. de thèse : M. Jin-Kao Hao Professeur à Université d’Angers
A CKNOWLEDGMENT
How time flies! I began my PhD study in Leria four years ago and I have finished all today. From a beginner who knows nothing about the optimization algorithms to a researcher who really enjoys the study of hard problems, I cannot express my thanks to all the people who helped me in just few lines.
I’m extremely grateful to my supervisor Prof. Jin-Kao Hao who provided me with encouragement and patience throughout the duration of my thesis. During the past four years, he gave me valuable guidance and advice not only on the research field but also on the learning attitude. I would like to extend my sincere thanks to my monitoring committee Prof. Eduardo Rodriguez-Tello who I cooperated a lot during the four years.
Thanks for his advice on the optimization methods and his kind invitation, it is sure that I will visit the pyramid in Mexico in the future.
I am also grateful to the jury members. Thanks for Prof. David Lesaint who manages my thesis defense and gave me much valuable advice in the future research. I’d also like to extend my gratitude to my referees Prof. Nicolas Durand and Prof. Yang Wang who reviewed my dissertation during this special period of Covid-19. I was quite enjoying the discussion with them and their detailed and difficult questions inspired me to do more theoretical work in the future.
I’d like to acknowledge the assistance of our technicians Eric Girardeau and Jean- Mathieu Chantrein, our nice secretaries Catherine Pawlonski and Christine Bardarine and all the other lab members. Thanks to them, I have enjoyed my life in Angers and had much fun in the daily life.
I am particularly grateful to my parents and my girlfriend. Thanks for all their love and selfless support! Without them, this work would not be possible. I also thank my Chinese friends Zequn Wei, Zhi Lu, Qing Zhou, Yinuo Li, Pengfei He and Zuocheng Li in Angers for their accompanies.
This research has been financially supported by China Scholarship Council (CSC).
3
T ABLE OF C ONTENTS
General Introduction 9
I Introduction 13
1 Introduction 15
1.1 Graph layout problems . . . 16
1.2 Applications . . . 17
1.3 Algorithms of the cyclic bandwidth problem and the minimum linear ar- rangement problem . . . 18
1.4 Algorithm assessment . . . 18
1.4.1 Benchmarks . . . 18
1.4.2 Metaheuristic algorithms evaluation . . . 19
II Contribution 21
2 An iterated three-phase search approach for solving the cyclic band- width problem 23 2.1 Introduction . . . 242.2 Iterated three-phase search for CBP . . . 28
2.2.1 Main scheme . . . 28
2.2.2 Extended evaluation function . . . 29
2.2.3 First phase - Double neighborhood descent search . . . 30
2.2.4 Second phase - Responsive threshold-based search . . . 35
2.2.5 Third phase - Shift-Insert-based perturbation . . . 37
2.3 Computational experiments . . . 39
2.3.1 Experimental setup . . . 39
2.3.2 Determination of the input parameter values for ITPS . . . 41
2.3.3 Comparison with the state-of-the-art algorithm . . . 42 5
2.4 Analysis . . . 44
2.4.1 Influence of the extended evaluation function. . . 46
2.4.2 Influence of the responsive threshold-based search . . . 47
2.4.3 Influence of the Shift-Insert-based perturbation . . . 48
2.5 Conclusions and future work . . . 49
3 A new iterated local search algorithm for the cyclic bandwidth problem 53 3.1 Introduction . . . 53
3.2 New iterated local search algorithm . . . 54
3.2.1 Dedicated tabu search . . . 55
3.2.2 Directed perturbation with randomized shift-insert . . . 58
3.2.3 Strong perturbation with destruction-reconstruction . . . 59
3.2.4 Relations with previous studies . . . 60
3.3 Experimental results . . . 62
3.3.1 Experimental setup . . . 62
3.3.2 Tuning of parameters . . . 63
3.3.3 Comparisons with state-of-the-art algorithms. . . 64
3.4 Analysis of the two perturbation strategies . . . 67
3.5 Conclusions . . . 70
4 A study of recombination operators for the cyclic bandwidth problem 73 4.1 Introduction . . . 73
4.2 Memetic Algorithm for CBP . . . 74
4.2.1 Search space, representation, fitness function . . . 74
4.2.2 General procedure . . . 76
4.2.3 Initialization. . . 76
4.2.4 Local search . . . 76
4.2.5 Recombination . . . 78
4.2.6 Updating population . . . 78
4.3 Recombination operators . . . 79
4.3.1 Order crossover . . . 79
4.3.2 Order-based crossover . . . 79
4.3.3 Cycle crossover . . . 80
4.3.4 Partially mapped crossover . . . 80
4.3.5 Distance preserved crossover . . . 81
TABLE OF CONTENTS
4.4 Experimental results . . . 81
4.4.1 Instances and settings . . . 81
4.4.2 Computational results . . . 81
4.5 Understanding the performance differences of the compared crossovers . . . 83
4.5.1 Distance and population entropy . . . 83
4.5.2 Interaction between crossover and problem characteristics . . . 84
4.6 Conclusions . . . 86
5 A set based neighborhood heuristic algorithm for solving the minimum linear arrangement problem 89 5.1 Introduction . . . 89
5.2 Previous work . . . 91
5.3 Set based neighborhood heuristic algorithm . . . 93
5.3.1 Representation and evaluation function . . . 94
5.3.2 Initialization . . . 94
5.3.3 Descent phase . . . 95
5.3.4 Perturbation phase . . . 100
5.4 Experimental results . . . 101
5.4.1 Instances and settings . . . 101
5.4.2 Determination of the input parameters . . . 102
5.4.3 Comparison with the other neighborhood algorithm . . . 103
5.5 Investigations of the key components . . . 107
5.5.1 Influence of the median based neighborhood . . . 107
5.5.2 Influence of the decomposition method . . . 109
5.6 Conclusions and perspectives. . . 109
III Conclusion 112
Conclusions 113
List of Figures 118
List of Tables 121
7
IV Appendix 122
6 Appendix 123
6.1 Detailed comparison of the IT P S and T Scb algorithms . . . 123 6.2 Detailed performance evaluation of N ILS with respect to the CBP state-
of-the-art algorithms . . . 123
List of Publications 133
Bibliography 135
G ENERAL I NTRODUCTION
Context
Graph layout problems are a particular class of combinatorial optimization problems.
Graph layout problems aim to find a linear layout of a given graph with respect to prede- fined optimization criteria. There are many problems which can be formulated as graph layout problems: bandwidth problem, cutwidth problem, sum cut problem, minimum lin- ear arrangement problem and so on ([DPS02]). In this thesis, we focus on two problems: the cyclic bandwidth problem (CBP) [LVW84] and the minimum linear arrangement problem (MinLA) [Har64]. As a variant of the bandwidth problem, CBP is to find an arrangement on a cycle to minimize the longest cyclic distance. MinLA is a min-sum problem whose goal is to find a linear arrangement of an input graph to minimize the sum of all edges.
The study of these two problems is meaningful for many fields such as very large-scale integration (VLSI) circuit design [BT84;Har64], bioinformatics [DPS02] and code designs [Chu88]. Given their importance, the research on these problems becomes more and more intense in recent years.
However, both problems are known to be N P-hard [Har82; Lin94]. It is unlikely to find the global optimal solution in a polynomial time unless P = N P. Even for small instances, exact algorithms would consume much time to find the optimal solution. But for most of their applications, feasible solutions of good quality are sufficient. Therefore, it is meaningful to develop effective heuristic and metaheuristic algorithms to find such solutions in a reasonable time. This thesis aims to develop effective heuristic algorithms for these problems. To assess the proposed algorithms, extensive experiments are carried out over available benchmarks and comparisons with state-of-the-art methods in the literature are shown. Moreover, we also investigate the key components of the proposed algorithms to reveal their influences on the performances of the algorithms.
Objectives
The main objectives of this thesis are summarized as follows:
9
— Understand deeply CBP and MinLA and analyse the main difficulty of developing heuristic algorithms for these problems.
— Explore effective and meaningful neighborhoods to operate efficiently in the local search procedure.
— Design problem-specific evaluation functions to help distinguish the solutions which have same objective value.
— Develop effective perturbation strategies to help escape from the local optimum traps.
— Investigate different recombination operators for the memetic algorithm (MA) to find out the feature for a suitable crossover.
— Evaluate the proposed algorithms over well-known benchmark instances and carry out a comprehensive comparison with the state-of-the-art algorithms.
Contributions
The main contributions of this thesis are summarized below:
— For CBP, we achieved the following results:
— Firstly, we proposed an iterated three-phase search approach (IT P S). The algorithm relies on three complementary search components to ensure a suit- able balance of search intensification and diversification, guided by an enriched evaluation function. Computational assessments on a test-suite of 113 popu- lar benchmark instances in the literature demonstrate the effectiveness of the proposed algorithm. This work has been published in IEEE Access.
— Secondly, we proposed a new iterated local search algorithm (N ILS). The algo- rithm relies on a simple, yet powerful local optimization procedure reinforced by two complementary perturbation strategies. The local optimization procedure discovers high-quality solutions in a particular search zone while the perturba- tion strategies help the search to escape from local optimum traps and explore unvisited areas. We present computational results on 113 benchmark instances from 8 different families, and show performances that are never achieved by current best algorithms in the literature. This study was presented in a paper submitted to Knowledge-Based Systems.
— Thirdly, we investigated five classical permutation crossovers within a basic memetic algorithm integrating a simple descent local search procedure. We
General Introduction
studied the correlation between algorithmic performances and population di- versity measured by the average population distance and entropy. This work has been selected by the conference Artificial Evolution 2019 and published in Lecture Notes in Computer Science.
— For MinLA, we studied a set based neighborhood heuristic algorithm under the framework of two phase iterated local search. The main contribution is the in- troduction of a set based neighborhood with a decomposition method in the de- scent phase. We compared the proposed algorithm with a traditional neighborhood heuristic algorithm, and computational results show that the set based neighbor- hood performs better than the traditional neighborhood in the local search phase.
Organization
The thesis is organized in the following way:
— In the first chapter, after introducing CBP and MinLA, we present related appli- cations and a brief overview of existing algorithms. The test benchmark and the computational assessments are shown at the end of the chapter.
— In the second chapter, we study the cyclic bandwidth problem and make a brief re- view of the existing algorithms. Then, we introduce the iterated three-phase search (IT P S) approach. We present experimental results of the proposed algorithm, as well as comparisons with state-of-the-art algorithms.
— In the third chapter, we propose another heuristic algorithm for CBP: the new iterated local search algorithm (N ILS). After introducing the simple but powerful neighborhood, the directed perturbation phase and special strong perturbation phases are presented. Then, experimental results ofN ILS and the state-of-the-art algorithms including IT P S and T Scbare shown over benchmark instances.
— In the fourth chapter, we focus on the study of different recombination operators for CBP. The general outline of the memetic algorithm for CBP is first described and 5 different crossovers are presented. Afterwards, computational results of the memetic algorithm with different crossovers are presented. Moreover, we investigate the correlation between the performance and the diversity of population.
— In the last chapter, we consider the minimum linear arrangement problem. We present the set based neighborhood with the decomposition method under the framework of two phase iterated heuristic algorithm. Comparative results with the
11
algorithm using the traditional 2-flip neighborhood as well as the state-of-the-art algorithm are presented.
Part I
Introduction
13
Chapter 1
I NTRODUCTION
15
1.1 Graph layout problems
Graph layout problems (GLP) are a class of combinatorial optimization problems.
Starting from the bandwidth minimization problem (BMP) [Har64] in the 1960s, GLP are to find a linear layout of an input graph in such a way that a certain objective function is optimized. Because of the strong application background, there are many theoretical and practical studies on GLP in the past decades. In this thesis, we mainly focus on two GLP: the cyclic bandwidth problem (CBP) and the minimum linear arrangement problem (MinLA).
— The cyclic bandwidth problem (CBP).
Let G(V, E) be a finite undirected graph of order n and Cn a cycle graph where V represents the set of vertices and E depicts the set of edges. Given a bijection ϕ:V →V which represents an embedding (also called a labeling) of GinCn, the cyclic bandwidth (the cost) for Gwith respect to ϕis defined as:
Cb(G, ϕ) = max
(u,v)∈E{|ϕ(u)−ϕ(v)|n}, (1.1)
where |x|n = min{|x|, n− |x|}, with |x| ∈ (0, n), is called the cyclic distance, and ϕ(u) denotes the label associated to vertex u. The cyclic bandwidth Cb(u, ϕ) for a particular vertex u, with set of adjacent vertices |A(u)| = deg(u), under the embeddingϕ can be computed as follows:
Cb(u, ϕ) = max
v∈A(u){|ϕ(u)−ϕ(v)|n}. (1.2)
The objective of CBP is thus to find an embedding ϕ∗ such that Cb(G, ϕ∗) is minimized:
ϕ∗ = arg min
ϕ∈Ω
{Cb(G, ϕ)}, (1.3)
where Ω represents the set of all possible embeddings. The embeddingϕ∗ satisfying this condition in Equation1.3 is called an optimal or exact embedding of the given graph.
— The minimum linear arrangement problem (MinLA).
Let G(V, E) be a finite undirected graph, where V (|V| = n) represents the set of vertices and E depicts the set of edges. Given a mapping ϕ : V → {1,2, ...n}
which represents a linear arrangement ϕ, the sum of edge length (the cost) for G
1.2. Applications
with respect to ϕis defined as:
SLA(G, ϕ) = X
(u,v)∈E
|ϕ(u)−ϕ(v)| (1.4)
The objective is to find an arrangement ϕ∗ whose the sum of total edge length SLA(G, ϕ∗) is minimal.
CBP and MinLA are bothN P-hard problems and their decision problems are known to beN P-complete [Har82; Lin94].
1.2 Applications
CBP has many applications in the real world. One of them is to design a ring inter- connection network [LVW84] for a couple of computers to ensure every message to be sent at its destination in less than certain steps. Another application is the VLSI design [BT84]. In the circuit layout domain, the maximum delay determines the clock-period of the system in the circuit. To increase the speed of the chip, it is essential to decrease the maximum delay by producing a layout with the longest edge as short as possible. Also, CBP is applied in data structure representations [RS78]. The replacement of the logical data structure with the physical storage structure is unavoidable step in the algorithm implementation procedure. This encoding of data structure could be seen as a match be- tween a “logical" guest structure and a “physical" host structure. An optimal solution for an input graph will offer a good match for the encoding which will improve the efficiency of the computers. Moreover, CBP is involved in the interconnection networks for parallel computer systems [Hro+92].
MinLA is a min-sum problem whose objective is to find a linear arrangement to mini- mize the sum of edge length for an input graph. It is widely used in many fields. First of all, the optimal solution of MinLA can derive the lower bound and upper bound for the bi- partite crossing number problem [Sha+00]. Concerning the design of the error-correcting codes [Har64], the application of MinLA will help minimize the average absolute error in message transmission. Also, in the field of VLSI, the vertices represent the pin on the chip while the edges depict the wire between the pin. The cost of the arrangement stands for the total wire length [AH73]. In addition, MinLA has other applications in biological fields, graph drawing, software diagram layout and job scheduling.
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1.3 Algorithms of the cyclic bandwidth problem and the minimum linear arrangement problem
The existing algorithms for CBP and MinLA fall into two categories: the exact algo- rithms and the heuristic algorithms.
For CBP, the majority of the studies focused on theoretical work: getting the exact value for certain family of graphs [LSC97;LSC02;Lin97] and finding lower bounds for the general graphs [CLS08;KNS11;Zho00]. There are few practical algorithms to solve CBP.
To our knowledge, there is only one exact algorithm, i.e. a branch and bound algorithm in [RRR12] which can solve small problem instances. Concerning the heuristic algorithms, a heuristic algorithm based on tabu search was presented in [Rod+15] to handle large and general instances.
For MinLA, there exist some exact methods to get the optimal solution for some special families of graphs such as trees, rooted trees, hypercubes, meshes, outerplanar graphs, and others ([DPS02]). Besides that, several heuristic algorithms were developed such as the spectral sequencing method (SSQ) [JM92], improved frontal increase mini- mization (IFIM) [Mca99], multi-scale algorithm (MS) [KH02], algebraic multigrid scheme (AMG) [SRB04], simulated annealing (SA) [Pet03a][Pet03b][RHT08a], population-based algorithms [RHT06] [Por05][SS09] and variable neighborhood search [MUP16].
The detailed review of previous work will be made for each considered problem in the following chapters.
1.4 Algorithm assessment
The assessment of the performance of an algorithm is based on experimental results over benchmark instances. In this section, we will make a brief introduction of the bench- mark instances and some indicators describing the algorithm performances.
1.4.1 Benchmarks
For CBP, the benchmark graphs are organized in two different groups.
— Standard Graphs. The first group is made up of 85 graphs belonging to 7 dif- ferent families of standard graphs (paths, cycles, two dimensional meshes, three dimensional meshes, complete r-level k-ary trees, caterpillars and r-dimensional
1.4. Algorithm assessment
hypercubes). Their order |V| varies in the range from 9 to 8192, while their size
|E| goes from 8 to 53248. The optimal solutions for these graphs are known, the reader is referred to [Rod+15] for the details. Therefore, attaining the optimal so- lutions for these instances is an important factor to evaluate the performance of algorithms.
— Harwell-Boeing Graphs.The second group contains 28 graphs from theHarwell- Boeing Sparse Matrix Collection.1 These instances were directly constructed from sparse adjacency matrices produced in practical and engineering real world ap- plications. Their order fluctuates in the interval 9 ≤ |V| ≤ 715 and their size is in the range 46 ≤ |E| ≤ 3720. The optimal solutions for 7 small graphs are al- ready known, while for the remaining 21 graphs lower and upper bounds can be calculated according to [Lin97].
For MinLA, the benchmark instances were introduced in [Pet03b]. There are 21 graphs which are classified into 5 groups.2.
— Random Graphs. This group consists of 5 graphs including 4 random graphs and a random geometric graph. The number of vertices of these graphs is 1000 and the number of edges is between 4974 and 49820.
— Regular Graphs. There are 3 graphs in this group: a complete binary tree with 10 levels, a 10-dimension hypercube and a 33×33 mesh graph. It is noticed that the optimal solutions of these graphs are known.
— FE Graphs. This group includes 3 graphs from finite element discretization. The numbers of vertices of these graphs are 4720, 4253 and 9800 respectively and the numbers of edges are 13722, 12289 and 28989 respectively.
— VLSI Graphs. This is a set of 5 graphs from the VLSI design. Their order fluc- tuates 828≤ |V| ≤1366 and their size are in the range 1749≤ |E| ≤2915.
— GD Graphs. It is composed of 5 graphs from graph drawing competitions. Four of them are in small size which have less than 180 vertices and 228 edges. And the other has 1096 vertices and 1676 edges.
1.4.2 Metaheuristic algorithms evaluation
To evaluate the performances of heuristic algorithms, the general method is to compare the results of different algorithms under the same environment over the same benchmark
1. They are downloadable at http://math.nist.gov/MatrixMarket/data/Harwell-Boeing 2. They are downloadable onhttps://www.tamps.cinvestav.mx/~ertello/minla.php
19
set. Due to the stochastic nature of heuristic algorithms, a common practice is to run the heuristic algorithm multiple times with different seeds (for example, we run our algorithms 50 times independently for each instance). Then we collect the results of each execution and establish the comparison between the proposed algorithms and the state-of-the-art algorithms. Normally, we consider the best objective value found, the average objective value for each instance, the running time to get the best value and the deviation of the solution quality. In this thesis, we also used other indicators such as the overall relative root mean square error to describe the overall performance of the algorithm. The sta- tistical significance test is implemented to verify whether the differences are statistically significant.
To shed light on the key components and reveal their influences to the performance of the proposed algorithm, additional control experiments are needed. Usually, we create several variants of the proposed algorithm by disabling or replacing some components.
The variants will follow the same experiment setting of the proposed algorithm over the benchmark set. This method is widely used in the following chapters and we will make specific presentation for each algorithm.
Part II
Contribution
21
Chapter 2
A N ITERATED THREE - PHASE SEARCH
APPROACH FOR SOLVING THE CYCLIC BANDWIDTH PROBLEM
In this chapter, we introduce an iterated three-phase search approach which relies on three complementary search components to ensure a suitable balance of search intensi- fication and diversification, guided by an enriched evaluation function. Computational assessments on a test-suite of 113 popular benchmark instances in the literature demon- strate the effectiveness of the proposed algorithm. In particular, it improves on 19 best- known computational results of the current best-performing algorithm for the problem and discovers 12 new record results (updated upper bounds). The key components of the proposed algorithm are investigated to shed light on their influences over the performance of the algorithm. The content of this chapter has been published in IEEE Access.
23
2.1 Introduction
The Cyclic Bandwidth Problem (CBP) is a general and useful model able to formulate a number of practical applications. Initially introduced in the context of designing ring interconnection networks [LVW84], CBP involves finding an arrangement on a cycle for a setV of computers with a known communication pattern, given by the graph G(V, E) to ensure that every message could be sent to its destination in at mostk steps. The decision problem of CBP is known to be N P-complete [Lin94]. Also, it has some other important applications in VLSI design [BT84], data structure representations [RS78], code designs [Chu88] and parallel computer systems [Hro+92].
Let G(V, E) be a finite undirected graph of order n and Cn a cycle graph. Given a bijection ϕ: V → V which represents an embedding (also called a labeling) of G inCn, the cyclic bandwidth (the cost) for G with respect to ϕis defined as:
Cb(G, ϕ) = max
(u,v)∈E{|ϕ(u)−ϕ(v)|n}, (2.1)
where |x|n = min{|x|, n− |x|}, with |x| ∈ (0, n), is called the cyclic distance, and ϕ(u) denotes the label associated to vertex u. The cyclic bandwidth Cb(u, ϕ) for a particular vertex u, with set of adjacent vertices |A(u)| = deg(u), under the embedding ϕ can be computed as follows:
Cb(u, ϕ) = max
v∈A(u){|ϕ(u)−ϕ(v)|n}. (2.2)
The main objective of CBP is thus to find an embedding ϕ∗ such that Cb(G, ϕ∗) is minimized:
ϕ∗ = arg min
ϕ∈Ω
{Cb(G, ϕ)}, (2.3)
where Ω represents the set of all possible embeddings. The embedding ϕ∗ satisfying this condition is called an optimal or exact embedding of the given graph.
Table 2.1 – Table of cyclic distances for all the edges of graph G depicted in Fig.2.1.
x= (u, v)∈E (a, b) (a, e) (a, f) (a, h) (b, d) (b, g) (c, e) (c, j) (d, i) (e, g) (f, g) (f, j) (h, i)
|x|=|ϕ(u)−ϕ(v)| 4 3 2 1 8 1 4 1 7 8 7 4 4
|x|n= min(|x|, n− |x|) 4 3 2 1 2 1 4 1 3 2 3 4 4
Fig.2.1shows a graph with ten vertices (n= 10) named fromatojwith an embedding
2.1. Introduction
or vertex labeling indicated in red from 1 to 10. In Fig.2.2, the vertices ofGare reordered clockwise on a cycle according to the label numbers (in red). So for each edge (u, v)∈E, it is easy to calculate the labels ϕ(u) and ϕ(v) to get the absolute distance |x| and the cyclic distance |x|n (see Table 2.1). For instance, the edge x = (f, g) (in blue) has an absolute distance |x| = 7 (i.e., the number of vertices from f to reach g in a clockwise direction in the cycle) while its cyclic distance|x|n equals 3 (min{7,10−7}, which is also the number of vertices from f to reach g in a counterclockwise direction). According to (2.1), the cyclic bandwidth Cb(G, ϕ) of this graph is the maximum value among all the
|x|n, i.e., Cb(G, ϕ) = 4 which concerns the edge x= (a, b) of Fig. 2.2 indicated in red.
Figure 2.1 – The vertices are named from a toj and a labeling is represented by the red numbers from 1 to 10.
Until now, the majority of the existing studies concern either special graphs whose exact cyclic bandwidths can be determined theoretically or propositions to define lower and upper bounds of a general graph. For instance, in [YZ95], it was shown that for every unit interval graph, there exists a simultaneously optimal labeling for several labeling problems including CBP. The study of [Hro+92] established the relationships between the bandwidth BP(G) and the cyclic bandwidth Cb(G): BP(G)≥Cb(G)≥ 12BP(G).
Following this result, studies of [LSC97; LSC02; Lin97] identified the criterion condi- tions for two extreme casesBP(G) = Cb(G) and 12BP(G) = Cb(G), and further obtained some exact values for special graphs including trees, planar graphs, triangulation meshes, grids with specific characteristics and some other graphs with particular conditions.
In [Zho00], a systematic method was proposed to achieve a number of lower bounds for the bandwidth of a graph, which is then used to obtain lower bounds for CBP in terms of some distance- and degree-related parameters.
Figure 2.2 – The graphGof Fig.2.1 with its verticesatoj reordered clockwise on a cycle according to the label numbers 1 to 10 (in red).
The work of [CLS08] was devoted to the upper bound of the cyclic bandwidth of a general graph with an edge added. By exploring the property that the cyclic distance between any pair of adjacent vertices will not be affected by shifting all vertices in the cyclic order the same distance, a sharp upper bound was obtained.
The study of [KNS11] used the semi-definite programming (SDP) relaxations of the quadratic assignment problem to propose two new lower bounds on the bandwidth and cyclic bandwidth, which are shown to be better than two other previous SDP bounds.
In addition to these theoretical results, little effort has been made to develop practical solution methods for CBP. To our knowledge, there are only three published algorithms on solving CBP. In [RRR12], a branch and bound algorithm was proposed that can solve some standard instances (like path, mesh and cycle) of small sizes limited to 40 vertices. To handle larger instances, a heuristic algorithm based on the tabu search meta- heuristic (named T Scb) was presented in [Rod+15]. The authors also adapted a highly effective simulated annealing designed for the related Bandwidth Minimization Problem (BMP) [Har64] to the cyclic bandwidth problem. Their experimental assessment on a set of benchmark instances demonstrated the superiority of T Scbover the simulated anneal- ing algorithm. As a result, T Scb can be considered as the state-of-the-art algorithm for CBP and will serve as the main reference for our computational study.
Our literature review indicates that contrary to the BMP, for which various solution
2.1. Introduction
methods have been proposed (e.g., [Mla+10; Piñ+04;RHT08b;Tor+15]), effective algo- rithms dedicated to CBP remains scarce. To enrich the practical solution arsenal for this important optimization problem, we introduce in this work an iterated three-phase search algorithm (IT P S) for solving CBP. The algorithm is characterized by the following fea- tures. First, the algorithm is composed of three key search components: a double neighbor descent phase to find a local optimal solution, a responsive threshold-based search phase to explore the nearby regions for the purpose of discovering better solutions and a special perturbation phase to displace the search to a new and distant region. The algorithm also integrates an extended evaluation function which enriches the optimization objective by additional information. This function is used to discriminate many solutions with the same cyclic bandwidth and provides a relevant means for guiding the search process.
We assess the proposed algorithm on a set of 113 well-known benchmark instances taken from the literature. This set of instances includes 85 standard graphs (e.g., paths, cycles, caterpillars, etc) and 28 Harwell-Boeing graphs which arise from diverse engineering and scientific real-world problems. The comparisons with the results produced by the state-of-the-art reference method show the competitiveness of our algorithm. For the set of 85 standard graphs, our algorithm improves on 19 best computational (upper) bounds and matches 60 best-known computational results from the literature. For the set of 28 Harwell-Boeing graphs, our algorithm discovers new record results (updated upper bounds) for 12 graphs and matches the best-known results for 15 other graphs.
The remainder of this chapter is organized as follows: Section 2.2 first introduces the main scheme of the proposed algorithm. Then, the implementation details of the neighbor- based descent procedure as well as the responsive threshold-based search method are pre- sented. In Section 2.3 a set of computational experiments is presented. They are devoted to determine the best input parameter values for theIT P S algorithm and to compare its performance with respect to the reference algorithm in the literature,T Scb [Rod+15]. Sec- tion2.4 experimentally investigates the extent to which key components of theIT P S al- gorithm can influence its global performance. Finally, the main conclusions drawn from this work in Section 2.5.
2.2 Iterated three-phase search for CBP
2.2.1 Main scheme
The proposed IT P S algorithm was inspired by the three-phase approach presented in [FH15]. Even if the work of [FH15] concerns a particular optimization problem (i.e., the quadratic minimum spanning tree problem), the approach is of general interest and has been applied to other problems such as clique partitioning [ZHG16]. In this work, we adapted this three-phase approach to CBP by reusing its general framework and making dedicated adaptations to deal with the particular features of our considered problem.
LetG= (V, E) be a graph of order|V|=nand a cycle graphCn = (V0, E0), the search space Ω considered by ourIT P S algorithm is composed of all candidate embeddings (la- bellings or solutions) of G in Cn, ϕ : V → V0. In our implementation, an embedding ϕ is represented by a permutation of {1,2, . . . , n} such that the i-th element denotes the label assigned to vertex i ∈ V. To effectively explore the space Ω, IT P S combines a double neighborhood descent search, a responsive threshold-based search as well as a specific perturbation. To cope with the difficulty of discriminating many equal-cost can- didate solutions,IT P S integrates an extended evaluation function using graph structure information.
The pseudo-code of the IT P S algorithm is presented in Algorithm 1. It starts with a randomly generated solution ϕ. Then the algorithm enters the main ‘while’ loop (lines Algorithm 1 IT P S algorithm for CBP
1: Input: Finite undirected graphG(V, E), neighborhoodsN1andN2, extended evaluation function fe, search depthδ and cutoff time limitTmax
2: Output: The best solution foundϕ∗ 3: ϕ←InitialSolution()
4: ϕ∗←ϕ
5: whilethe cutoff time limit Tmax is not reacheddo 6: N onImp←0
7: while N onImp < δ do
8: (ϕ, ϕ∗)←DN DS(ϕ, ϕ∗, N1, N2) // Section2.2.3
9: (ϕ, ϕ∗)←RT BS(ϕ, ϕ∗, N1, N2) // Section2.2.4
10: N onImp←N onImp+ 1 11: end while
12: ϕ←P urturbation(ϕ) // Section2.2.5
13: end while 14: return ϕ∗
2.2. Iterated three-phase search for CBP
5-13), Alg.1) to explore solutions of increasing quality in terms of the extended evaluation function fe. At each iteration, the descent search (first phase, Section 2.2.3) is first run to find a local optimal solution using two neighborhoodsN1 and N2 (line 8, Alg.1). This phase is followed by the responsive threshold-based search (second phase, Section 2.2.4) to discover additional local optima of better quality from the incumbent solution (line 9, Alg.1). These two phases are repeated δ times. At this point, the search is judged to be trapped in a deep local optimum. To overcome the trap, the perturbation procedure (third phase, Section2.2.5) is triggered to strongly transform the incumbent solution ( line 12, Alg. 1). The search then goes back to the first phase with the perturbed solution as its new starting solution. During the search, each time a solution better than the previous best recorded solution is found, ϕ∗ is updated. The whole search process stops when a given cutoff time limit (Tmax) is reached. As the output of the algorithm, the best recorded solutionϕ∗ is returned.
2.2.2 Extended evaluation function
A notable feature of CBP is that many solutions may have the same objective value.
This is because there are (n−1)!/2 possible solutions while there are onlybn/2cdifferent possible objective values, see equation (2.1). From the local optimization perspective, it is critical to discriminate the solutions with the same objective value. For this purpose, we devise an extended evaluation function fe as follows.
Letϕ∈Ω be a candidate solution with cyclic bandwidth cost Cb(G, ϕ). LetN umE(Cb(G, ϕ)) represent the number of edges whose cyclic bandwidth equals Cb(G, ϕ):
N umE(Cb(G, ϕ)) = X
(u,v)∈E
Xuv, (2.4)
where Xuv = 1 if |ϕ(u)−ϕ(v)|n = Cb(G, ϕ); otherwise Xuv = 0. Then, the extended evaluation function fe is given by:
fe(ϕ) = Cb(G, ϕ) + N umE(Cb(G, ϕ))
|E| . (2.5)
As we show below, this evaluation function is able to distinguish the solutions that un- der the conventional evaluation function presented in (2.1) have the same objective value.
An analysis of the influence of the new evaluation functionfe is provided in Section2.4.1.
Figure2.3 shows an example of the extend evaluation functionfe applied to two solu-
(a) (b)
Figure 2.3 – An illustration of the extended evaluation functionfeapplied to two different embeddings. (a)ϕ1. (b)ϕ2. Both embeddings have the same cost (cyclic bandwidth) under the conventional evaluation function (2.1). However, the new function fe discriminates these embeddings by assigning to them two different values fe(ϕ1) = 4 + 1/13 = 4.0769 and fe(ϕ2) = 4 + 3/13 = 4.2307.
tions with the same objective value Cb(G, ϕ) = 4. According to the extended evaluation function, fe(ϕ1) = 4 + 1/13 = 4.0769, whilefe(ϕ2) = 4 + 4/13 = 4.3076. The embedding ϕ1 is considered to be “better” than the embedding ϕ2. This is reasonable, because one notices, from Fig.2.3(a), that for reducing the cost value Cb(G, ϕ) of the embeddingϕ1 it is necessary to deal with only one edge (marked in red), while for embeddingϕ2, depicted in Fig. 2.3(b), there are four edges (marked in red) that should be considered. Thus, it is easier to operate with ϕ1 than with ϕ2 to reduce the cyclic bandwidth of G.
2.2.3 First phase - Double neighborhood descent search
To explore the given search space, we first apply the double neighborhood descent search procedure (DNDS) whose general scheme is shown in Algorithm2. Basically, DNDS explores the two neighborhoods N1 and N2 defined below and iteratively replaces the incumbent solution by a neighbor solution selected from a set of candidate neighbors.
At each iteration, DNDS uses either N1 or N2 to create the candidate list (CLst) by identifying the solutions no worse than the incumbent solution in terms of the evaluation functionfe (lines 6-16, Alg.2). A priority is always given toN1 andN2 is examined only if the neighbor solutions inN1are all worse than the incumbent solution. If the candidate list
2.2. Iterated three-phase search for CBP
Algorithm 2 Double neighborhood descent search
1: Input: input solutionϕ, best optimum foundϕ∗, neighborhoodsN1andN2, evaluation function fe, maximum non-improving limitLd, and best neighbor move strategy probabilityρbest
2: Output:last local optimum ϕ, best optimum foundϕ∗ 3: N onImpCounter←0
4: Improving←T rue
5: while N onImpCounter < Ld do 6: if Improving then
7: N ←N1
8: else
9: N ←N2
10: end if 11: CLst← ∅
12: for eachϕ0∈N(ϕ) do 13: if fe(ϕ0)≤fe(ϕ)then 14: CLst←CLst∪ {ϕ0} 15: end if
16: end for
17: if CLst6=∅then
18: if rand(0,1)< ρbest then
19: ϕ←BestSol(CLst) // With probability ρbest
20: else
21: ϕ←RandomSol(CLst) 22: end if
23: Improving←T rue 24: else
25: Improving←F alse 26: end if
27: if fe(ϕ)< fe(ϕ∗) then 28: N onImpCounter←0
29: ϕ∗ ←ϕ
30: else
31: N onImpCounter←N onImpCounter+ 1 32: end if
33: end while 34: return ϕ,ϕ∗
is not empty (i.e., it contains at least one improving or non-worsening neighbor solution), either one best neighbor solution, or a random neighbor solution is chosen from CLst to become the new current solution according to probability ρbest (lines 18-22, Alg. 2).
Notice that given the criterion used to buildCLst, the selected neighbor solution is always at least as good as the replaced solution. In case CLst contains no candidate solution,
DNDS moves to the next iteration without performing a solution transition (the number of consecutive non-improving iterations is indicated byN onImpCounter, line 31, Alg.2).
During the search, the best-found solution ϕ∗ is updated each time a better solution is attained. The DNDS process terminates if the best-found solution ϕ∗ cannot be updated duringLdconsecutive iterations. In this case, DNDS has attained a local optimum and the IT P S algorithm switches to the responsive threshold-based search method for escaping this local optimum trap and to continue looking for new better quality solutions.
Neighborhoods
The two neighborhoodsN1andN2 explored by DNDS are defined by the generalswap operator. Let ϕbe the incumbent solution, then a neighbor solution ϕ0 can be generated by exchanging the labels of vertices u and v with the operation swap(u, v). Without any restriction, theswapoperator leads to a neighborhood of size of orderO(n2), which is too large to be explored efficiently. Following the idea of [Rod+15], we use two constrained neighborhoods by imposing specific conditions on the vertices that take part in a swap operation.
The first neighborhood N1(ϕ) is given by the set of neighbor solutions obtained by swapping a critical vertex u∈C(ϕ) and a specific vertex v ∈S(u) adjacent tou:
N1(ϕ) ={ϕ0 =ϕ⊕swap(u, v) :u∈C(ϕ), v ∈S(u), swap(u, v)∈/ T L},
(2.6)
whereϕ0 =ϕ⊕swap(u, v) denotes the neighbor solution obtained by applyingswap(u, v) to transform ϕ, T L is the so-called tabu list that records the swaps that were recently performed (see Section 2.2.3). The set C(ϕ) contains a group of critical vertices w ∈ V having a cyclic bandwidth Cb(w, ϕ) = Cb(G, ϕ), while S(u) ⊆ V is the set containing those verticeszcurrently labeled with values closer tomid(u) than toϕ(u) (i.e.,|mid(u)−
ϕ(z)|n <|mid(u)−ϕ(u)|n). The value mid(u) stands for the middle point of the shortest path in the cycle Cn containing all the vertices adjacent to u [Rod+15].
The descent procedure uses this strongly constrained neighborhoodN1(ϕ) to make an intensified exploration of candidate solutions.
Figure 2.4 depicts an illustrative example of the neighborhood N1(ϕ). It presents an embeddingϕcontaining a critical vertex c∈C(ϕ) (marked in red), which has the label 6
2.2. Iterated three-phase search for CBP
Figure 2.4 – A simple illustration of the neighborhoodN1(ϕ). The embeddingϕcontaining a critical vertex c∈C(ϕ) (marked in red), as well as the set S(c) ={a, h, f} of suitable vertices eligible to be swapped with vertexc (highlighted in blue) are depicted.
assigned to it. Using its adjacent verticesA(c) = {e, j}(edge (c, e) and edge (c, j) marked in green), we identify the vertex h (having label 4) as the middle point mid(c) of the shortest path in the cycle Cn containing all the vertices in A(c). Thus, all the vertices highlighted in blue (i.e., a, h and f) are in the suitable set S(c) and are eligible to be swapped with vertexc.
For the purpose of search diversification, the descent procedure employs also a larger neighborhood N2(ϕ) which is specified by the following expression:
N2(ϕ) ={ϕ0 =ϕ⊕swap(u, v) :u∈C(ϕ), v ∈Rγ(u), swap(u, v)∈/ T L},
(2.7)
where the setRγ(u)⊆V containsγ∗nrandomly selected vertices (γ ∈(0,1]). Compared to N1(ϕ), the swap operator can exchange a critical vertex u with any other vertex in the graph, leading to a much higher freedom for a swap operation. Since the neighbor solutions of N2(ϕ) are more varied, this neighborhood promotes search diversification.
Compared to swapping all pairs of labels to generate neighbor solutions, the neighbor- hoods N1(ϕ) and N2(ϕ) are much smaller in size. Indeed, N1(ϕ) contains |C(ϕ)| ∗ |S(.)|
neighbor solutions, where |S(.)| is the average number of suitable vertices for a critical vertex with respect to the current solution ϕ, while N2(ϕ) has |C(ϕ)| ∗γ ∗n neighbor solutions.
Our preliminary experiments indicated that for the tested instances |C(ϕ)| ≤ 0.1∗n and |S(.)| ≤ 0.1∗n hold. For this reason the value of γ was set to 0.05 or 0.1 in our experiments. As a result, each iteration of the descent procedure only considers 0.01n2 candidate solutions, which significantly accelerates the search process.
Finally, we adopted a fast incremental technique to evaluate a neighbor solution ϕ0 according to the evaluation function fe. Let ϕ0 be an embedding obtained by swapping u and v in ϕ. Then, to obtain fe(ϕ0) from fe(ϕ), we need only to recalculate the changing part |A(u)|+|A(v)| (|A(u)| and |A(v)| represent the number of adjacent vertices to u and v, respectively). This ensures that each iteration of the algorithm requires a time complexity bounded by O((|A(u)|+|A(v)|)∗n2).
Tabu list management
Since the double neighborhood descent search only accepts non-deteriorating (i.e., improving or equal cost) neighbor solutions, it is possible that a previously visited solution is reconsidered at a later iteration, leading to search cycling. To avoid this problem, the DNDS procedure integrates a tabu list that is a key concept of the tabu search method [GL97]. The idea is to keep track of the performed swaps and forbid the reverse swap operations during the next τ iterations (τ is an input parameter called the tabu tenure).
So when swap(u, v) is performed to transform the current solution, swap(u, v) is added in the tabu list and it is forbidden to swap verticesv anduduring the period fixed by the tabu tenure. In principle, the tabu tenure can take a fixed value or can be dynamically calculated during the search. We adopt a dynamic tabu tenure technique introduced in [GBF11]. As shown in other studies [LH16; WH13], this technique proves to be robust and effective in different settings and was also used in [Rod+15] for CBP. This technique applies a periodic step function that takes as argument the number of iterations iter for computing the tabu tenure value. The value returned by this function for a particular iteration iter is given by (aj)j=1,2,...,15 = (1,2,1,4,1,2,1,8,1,2,1,4,1,2,1)×d, where d is a parameter fixing the minimum tabu tenure (set to 100 in this work) and index j is computed by j =biter mod 1500100 c+ 1. Therefore, each period of this function is composed of 1500 iterations divided into 15 intervals.
2.2. Iterated three-phase search for CBP
Discussions
Like [Rod+15], the first phase of our ITPS algorithm is based on two neighborhoods.
However, there are some notable differences. First, our neighborhood N1 uses a set C(ϕ) of critical vertices defined by the condition Cb(w, ϕ) = Cb(G, ϕ), which is more restrictive than the condition Cb(w, ϕ) ≥ α∗Cb(G, ϕ) (α is a prefixed parameter between 0 and 1) used in [Rod+15]. In this way, the set of critical vertices is reduced and each iteration needs to examine fewer candidate solutions. Second, we make a swap move after visiting all candidate solutions induced by all critical vertices inC(ϕ) while in [Rod+15] a swap move is performed after visiting the candidate solutions of only one critical vertex. The advantage of our strategy is that we could encounter a better solution at each iteration, and have less chance to miss an elite solution. Third, in [Rod+15], the two neighborhoods are used according to a probability. In our work,N1 is always applied with priority andN2 is used only whenN1is exhausted (i.e., when a local optimum is attained withN1). Finally, our first phase uses the descent procedure to ensure an efficient search intensification (i.e., no worsening neighbor solution is allowed), while the algorithm of [Rod+15] uses tabu search which may accept worsening solution transitions.
2.2.4 Second phase - Responsive threshold-based search
As explained in Section 2.2.3, the double neighborhood based descent search only accepts non-deteriorating neighbor solutions. As such, it can be trapped in local optima.
When this happens, we trigger the second search phase and apply the responsive threshold- based search (RTBS) to escape such traps. During the second phase, both improving and deteriorating neighbor solutions can be accepted in order to favor a large exploration of the search space.
Like the double neighborhood based descent search, the responsive threshold-based search also relies on the neighborhoodsN1 and N2. However, RTBS adopts the threshold accepting heuristic [Due93; DS90] as the criterion for solution transitions. As such, a solution whose quality does not drop below a given threshold can be accepted to replace the incumbent solution. To further enforce search exploration, the two neighborhoods are considered alternatively according to a probabilityρN1. The general responsive threshold- based search procedure is described in Algorithm3.
RTBS starts each iteration by calculating the responsive threshold, denoted byT (line 5, Alg. 3). Then it iteratively makes transitions from the current solution to a neighbor
Algorithm 3 Responsive threshold-based descent procedures
1: Input: input solutionϕ, best found solutionϕ∗, neighborhoods N1 andN2, evaluation function fe, maximum non-improving limit Lt, neighborhood N1 application probability ρN1, and best neighbor move strategy probabilityρbest
2: Output: best found solutionϕ∗, last solutionϕ 3: N onImpCounterT ←0
4: whileN onImpCounterT < Lt do 5: T ←T hreshold(ϕ)
6: if rand(0,1)< ρN1 then
7: N ←N1 // With probability ρN1
8: else
9: N ←N2
10: end if 11: CLst← ∅
12: for eachϕ0 ∈N(ϕ) do 13: if fe(ϕ0)≤T then 14: CLst←CLst∪ {ϕ0} 15: end if
16: end for
17: if CLstis not empty then 18: if rand(0,1)< ρbest then
19: ϕ←BestSol(CLst) // With probabilityρbest
20: else
21: ϕ←RandomSol(CLst) 22: end if
23: end if
24: if fe(ϕ)< fe(ϕ∗) then 25: N onImpCounterT ←0
26: ϕ∗←ϕ
27: else
28: N onImpCounterT ←N onImpCounterT + 1 29: end if
30: end while 31: return ϕ,ϕ∗
2.2. Iterated three-phase search for CBP
solution which is selected by examining the neighborhoods N1 and N2. The former is applied with probability ρN1, while the latter is employed at a (1−ρN1) rate (lines 6-10, Alg. 3). This is simulated with a random number generated in the interval (0,1). Then all neighbor solutions whose quality is no worse than the threshold T are identified to form the CLst (lines 12-16, Alg. 3). Finally, according to the probability ρbest, either a best solution or a random solution is selected from CLst (like DNDS does) and used to replace the current solution (lines 18-22, Alg. 3). The best solution found ϕ∗ during the search is updated each time a better solution is discovered (lines 24-29, Alg. 3). If ϕ∗ is not updated, we increase the counter of non-improving iterationsN onImpCounterT and move to the next iteration. This process stops if the best local optimum found during this run can not be updated during Lt consecutive iterations. In this case, the search is supposed to be trapped in a deep local optimum.
One key issue concerns the threshold T. Indeed, if T takes a value that is far from the current objective value (T − Cb(G, ϕ)0), even very bad neighbor solutions can be accepted, leading to a random-like search. On the other hand, if T takes a value that is too close to the current objective value (T − Cb(G, ϕ) ≈ 0), the search will behave like the descent search and can hardly escape local optimum traps. To iden- tify a suitable threshold T, we follow the work of [CH15] and use a responsive mech- anism to dynamically tune T according to the current objective value Cb(G, ϕ) and a threshold ratio r. Specifically, we set T as follows T = (1 + r) ∗ Cb(G, ϕ), where r = 1/(a ∗ Cb(G, ϕ) + b) + c. The coefficients a, b, and c were empirically fixed at 0.00891104, 0.52663736 and 0.16331589, respectively. It was carried out by solving si- multaneously three equations produced with the following pairs of (Cb(G, ϕ), r) values obtained from preliminary experiments: {(2,2),(150,0.7),(3000,0.2)}. As a result, the threshold T evolves according to Cb(G, ϕ) and the threshold ratio r. T tends to become small when the current solution is of high quality so that only improving or limited wors- ening neighbor solutions are accepted. Inversely,T tends to become large when the current solution is not so good in order to encourage more exploration.
2.2.5 Third phase - Shift-Insert-based perturbation
With its threshold accepting strategy, the responsive threshold-based search ensures a large exploration of solutions of various quality. When this second phase is exhausted, we trigger a strong perturbation to displace the search to a new and distant region of the search space. Specifically, this is achieved by applying the Shif tInsert operator to
transform the current solution as follows.
Let ϕbe the current solution with cyclic bandwidth Cb(G, ϕ). LetW ={(u, v)∈E :
|ϕ(u)−ϕ(v)|n= Cb(G, ϕ)}be the set of edges whose cyclic distance equals Cb(G, ϕ). Let e= (u, v) be an edge randomly taken fromW such thatϕ(u)> ϕ(v). TheShif tInsert(u, v) operator first removesu, then shifts all vertices betweenuandv clockwise or anti-clockwise at random, and finally insertsu at the position ofv. In practice,Shif tInsert(u, v) is re- alized by performing Cb(G, ϕ)−1 successive swap(u, x) operations where x denotes the inverse clockwise nearby vertex of u in the solution undergoing transformation until x reaches vertex v.
An illustrative example is shown in Fig. 2.5(a) (solution before the Shif tInsert op- eration) and Fig. 2.5(b) (solution after the Shif tInsert operation). In this example, Cb(G, ϕ) = 4 and edge (c, e) is chosen forShif tInsertamongW ={(c, e),(f, j),(h, i),(a, b)}, which is the set of edges with a cyclic distance of 4. Shif tInsert(c, e) is performed by three successive swap operations: swap(c, a), swap(c, h), and swap(c, f). Table 2.2 indi- cates the changes of the cyclic distances of the edges impacted by the Shif tInsert(c, e) operation.
The Shift-Insert-based perturbation has some interesting features. On the one hand, by displacing a significant number of vertices, this strategy helps to break long standing ties and forces the search to overcome deep local traps. Second, by considering edges whose cyclic distance is equal to the current cyclic bandwidth, this strategy maintains the quality of the transformed solution at a reasonable level and thus avoids searching from a lower quality solution.
When the third phase is triggered, the Shift-Insert-based perturbation is applied one time to transform the current solution. The modified solution is then used as the new starting solution of the next round of theIT P S algorithm. In Section2.4.3, we investigate the usefulness of the Shift-Insert-based perturbation.
Table 2.2 – Changes of the cyclic distances associated to the edges impacted by the Shif tInsert operation when applied over the solution depicted in Fig.2.5.
(u, v) (f, g) (f, j) (h, i) (h, a) (a, f) (a, e) (a, b)
|ϕ(u)−ϕ(v)|n 3 4 4 1 2 3 4
|ϕ(u)−ϕ(v)|n 4 3 3 1 2 4 3
Change +1 -1 -1 0 0 +1 -1
2.3. Computational experiments
(a) (b)
Figure 2.5 – An illustrative example of the Shift-Insert-based perturbation. (a) Solution ϕ before applying the Shif tInsert perturbation. (b) Solution ϕ after applying the per- turbation Shif tInsert(c, e).
2.3 Computational experiments
This section is dedicated to an experimental assessment of the proposed IT P S algo- rithm, the experimental setup, the test-suite, the procedure used to set the parameter values and a performance comparison between IT P S and T Scb (the reference state-of- the-art method) [Rod+15].
2.3.1 Experimental setup
The IT P S algorithm described in the previous section was coded in the C++ pro- gramming language1. We have also the C source code of the T Scb algorithm2. Thus, both algorithms were compiled with g++ version 4.4.7 using the optimization flag -O3.
All the experiments presented in this work were run sequentially on the same computa- tional platform with a CPU Intel Xeon X5650 at 2.66 GHz, 2 GB of RAM with Linux operating system. For each benchmark instance a total of 50 independent executions, us- ing different random seeds, of the analyzed algorithms were accomplished due to their stochastic nature.
1. The source code of ourIT P S algorithm is available at: https://github.com/thetopjiji/ITPS 2. The source code of theT Scbalgorithm reported in [Rod+15] is available at:https://www.tamps.
cinvestav.mx/~ertello/cbmp.php
